Local degree at an isolated preimage

Let $f:S^n\to S^n$ be continuous, $n\ge 1$, with oriented source and target as in Degree of a self map of an oriented sphere. Suppose $y=f(x)$ and $x$ is isolated in $f^{-1}(y)$. Choose an open neighborhood $U$ of $x$ with $U\cap f^{-1}(y)=\{x\}$. The map of pairs $(U,U\setminus \{x\})⟶(S^n,S^n\setminus \{y\})$ induces a homomorphism between infinite cyclic groups. Its integer multiplier in the generators restricted from the two global orientation classes is the local degree ${\mathrm{deg}}_{x}f$.

Excision Excision for singular homology identifies the domain local group with $H_n(S^n,S^n\setminus \{x\};\mathbb{Z})$; the pair sequence Long exact sequence of a pair supplies the global-to-local identification. The following lemma establishes these identifications and independence of the neighborhood.